Subgroup Structure of Interval Exchange Transformations

Definitions and mathematical statement

Under the usual orientation-preserving convention, an interval exchange transformation is a bijection $T:[0,1)\to[0,1)$ for which there is a finite partition into half-open intervals $I_1,\ldots,I_r$ such that $T|_{I_i}$ is a translation and the image intervals partition $[0,1)$. These transformations form a group $G$ under composition. This definition fixes a conventional reading; the extraction itself does not specify whether transformations with flips, endpoint identifications, or other variants are included.

The source asks two broad questions: describe the group-theoretic structure of $G$, and characterize the abstract groups $H$ for which there exists an injective homomorphism $H\to G$. No finite-generation, countability or other restriction on $H$ is stated. The structure question does not name a particular invariant or proposed property, and the subgroup question does not supply a conjectured classification. Accordingly these are retained as research directions, not converted into an invented universal assertion about amenability, free subgroups, or another property.

Short English statement

Study the algebraic structure of the interval-exchange group and determine which abstract groups embed in it. The conventional no-flips definition is explained, while the source's broad classification scope and missing variant conventions remain explicit.

Sources

• Local UnsolvedMath registry, permanent ID $30001037$, source code catalogue entry, "Subgroup Structure of Interval Exchange Transformations." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, pp. 2337--2406, DOI https://doi.org/10.4171/owr/2008/42.
• Supplied "Dahmani--Fujiwara--Guirardel": https://arxiv.org/abs/1101.5909.
• Supplied "Dahmani, survey": https://numdam.org/articles/10.5802/wbln.27/.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled open; says primary and survey literature provides substantial subgroup obstructions but no full structural or subgroup classification resolving these questions. This is supplied dated curation, not independently verified.